Fundamental solutions in elastodynamics: a compendium
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
New York
Cambridge University Press
2006
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Table of contents Publisher description Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references |
| Physical Description: | X, 251 S. graph. Darst. 27 cm |
| ISBN: | 9780521855709 0521855705 |
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| 245 | 1 | 0 | |a Fundamental solutions in elastodynamics |b a compendium |c Eduardo Kausel |
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| adam_text | FUNDAMENTAL SOLUTIONS IN ELASTODYNAMICS A COMPENDIUM EDUARDO KAUSEL
MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE UNIVERSITY PRESS
CONTENTS PREFACE PAGE IX SECTION I: PRELIMINARIES 1. FUNDAMENTALS 1 1.1
NOTATION AND TABLE OF SYMBOLS 1 1.2 SIGN CONVENTION 4 1.3 COORDINATE
SYSTEMS AND DIFFERENTIAL OPERATORS 4 1.3.1 CARTESIAN COORDINATES 4 1.3.2
CYLINDRICAL COORDINATES 6 1.3.3 SPHERICAL COORDINATES 9 1.4 STRAINS,
STRESSES, AND THE ELASTIC WAVE EQUATION 13 1.4.1 CARTESIAN COORDINATES
13 1.4.2 CYLINDRICAL COORDINATES 16 1.4.3 SPHERICAL COORDINATES 21 2.
DIPOLES 27 2.1 POINT DIPOLES OR DOUBLETS: SINGLE COUPLES AND TENSILE
CRACK SOURCES 27 2.2 LINE DIPOLES 29 2.3 TORSIONAL POINT SOURCES 30 2.4
SEISMIC MOMENTS (DOUBLE COUPLES WITH NO NET RESULTANT) 30 2.5 BLAST
LOADS (EXPLOSIVE LINE AND POINT SOURCES) 31 2.6 DIPOLES IN CYLINDRICAL
COORDINATES 32 SECTION II: FULL SPACE PROBLEMS 3. TWO-DIMENSIONAL
PROBLEMS IN FULL, HOMOGENEOUS SPACES 35 3.1 FUNDAMENTAL IDENTITIES AND
DEFINITIONS 35 3.2 ANTI-PLANE LINE LOAD (SH WAVES) 35 3.3 SH LINE LOAD
IN AN ORTHOTROPIC SPACE 36 3.4 IN-PLANE LINE LOAD (SV-P WAVES) 38 3.6
LINE BLAST SOURCE: SUDDENLY APPLIED PRESSURE 43 3.7 CYLINDRICAL CAVITY
SUBJECTED TO PULSATING PRESSURE 44 4. THREE-DIMENSIONAL PROBLEMS IN
FULL, HOMOGENEOUS SPACES 48 4.1 FUNDAMENTAL IDENTITIES AND DEFINITIONS
48 4.2 POINT LOAD (STOKES PROBLEM) 48 4.3 TENSION CRACKS 52 4.4 DOUBLE
COUPLES (SEISMIC MOMENTS) 54 4.5 TORSFONAL POINT SOURCE 55 4.6 TORSIONAL
POINT SOURCE WITH VERTICAL AXIS 57 4.7 POINT BLAST SOURCE 58 4.8
SPHERICAL CAVITY SUBJECTED TO ARBITRARY PRESSURE 59 4.9 SPATIALLY
HARMONIC LINE SOURCE (2V 2 -D PROBLEM) 63 SECTION III: HALF-SPACE
PROBLEMS 5. TWO-DIMENSIONAL PROBLEMS IN HOMOGENEOUS HALF-SPACES 69 5.1
HALF-PLANE, SH LINE SOURCE AND RECEIVER ANYWHERE 69 5.2 SH LINE LOAD IN
AN ORTHOTROPIC HALF-PLANE 70 5.3 HALF-PLANE, SV-P SOURCE AND RECEIVER AT
SURFACE (LAMB S PROBLEM) 71 5.4 HALF-PLANE, SV-P SOURCE ON SURFACE,
RECEIVER AT INTERIOR, OR VICE VERSA 73 5.5 HALF-PLANE, LINE BLAST LOAD
APPLIED IN ITS INTERIOR (GARVIN S PROBLEM) 76 6. THREE-DIMENSIONAL
PROBLEMS IN HOMOGENEOUS HALF-SPACES 78 6.1 3-D HALF-SPACE, SUDDENLY
APPLIED VERTICAL POINT LOAD ON ITS SURFACE (PEKERIS-MOONEY S PROBLEM) 78
6.2 3-D HALF-SPACE, SUDDENLY APPLIED HORIZONTAL POINT LOAD ON ITS
SURFACE (CHAO S PROBLEM) 81 6.3 3-D HALF-SPACE, BURIED TORSIONAL POINT
SOURCE WITH VERTICAL AXIS 83 SECTION IV: PLATES AND STRATA 7.
TWO-DIMENSIONAL PROBLEMS IN HOMOGENEOUS PLATES AND STRATA 87 7.1 PLATE
SUBJECTED TO SH LINE SOURCE 87 7.1.1 SOLUTION USING THE METHOD OF IMAGES
87 7.1.2 NORMAL MODE SOLUTION 87 7.2 STRATUM SUBJECTED TO SH LINE SOURCE
88 7.2.1 SOLUTION USING THE METHOD OF IMAGES 89 7.2.2 NORMAL MODE
SOLUTION 90 7.3 PLATE WITH MIXED BOUNDARY CONDITIONS SUBJECTED TO SV-P
LINE SOURCE 90 7.3.1 SOLUTION USING THE METHOD OF IMAGES 91 7.3.2 NORMAL
MODE SOLUTION 92 CONTENTS VII SECTION V: ANALYTICAL AND NUMERICAL
METHODS READ ME FIRST 97 8. SOLUTION TO THE HELMHOLTZ AND WAVE EQUATIONS
98 8.1 SUMMARY OF RESULTS 98 8.2 SCALAR HELMHOLTZ EQUATION IN CARTESIAN
COORDINATES 101 8.3 VECTOR HELMHOLTZ EQUATION IN CARTESIAN COORDINATES
102 8.4 ELASTIC WAVE EQUATION IN CARTESIAN COORDINATES 104 8.4.1
HORIZONTALLY STRATIFIED MEDIA, PLANE STRAIN 105 8.5 SCALAR HELMHOLTZ
EQUATION IN CYLINDRICAL COORDINATES 108 8.6 VECTOR HELMHOLTZ EQUATION IN
CYLINDRICAL COORDINATES 109 8.7 ELASTIC WAVE EQUATION IN CYLINDRICAL
COORDINATES 110 8.7.1 HORIZONTALLY STRATIFIED MEDIA 112 8.7.2 RADIALLY
STRATIFIED MEDIA 114 8.8 SCALAR HELMHOLTZ EQUATION IN SPHERICAL
COORDINATES 117 8.9 VECTOR HELMHOLTZ EQUATION IN SPHERICAL COORDINATES
118 8.10 ELASTIC WAVE EQUATION IN SPHERICAL COORDINATES 120 9. INTEGRAL
TRANSFORM METHOD 125 9.1 CARTESIAN COORDINATES 125 9.2 CYLINDRICAL
COORDINATES 130 9.2.1 HORIZONTALLY STRATIFIED MEDIA 130 9.2.2
CYLINDRICALLY STRATIFIED MEDIA 136 9.3 SPHERICAL COORDINATES 137 10.
STIFFNESS MATRIX METHOD FOR LAYERED MEDIA 140 10.1 SUMMARY OF METHOD 141
10.2 STIFFNESS MATRIX METHOD IN CARTESIAN COORDINATES 142 10.2.1
ANALYTIC CONTINUATION IN THE LAYERS 148 10.2.2 NUMERICAL COMPUTATION OF
STIFFNESS MATRICES 150 10.2.3 SUMMARY OF COMPUTATION 151 10.3 STIFFNESS
MATRIX METHOD IN CYLINDRICAL COORDINATES 159 10.3.1 HORIZONTALLY LAYERED
SYSTEM 160 10.3.2 RADIALLY LAYERED SYSTEM 164 10.4 STIFFNESS MATRIX
METHOD FOR LAYERED SPHERES 175 10.4.1 PROPERTIES AND USE OF IMPEDANCE
MATRICES 180 10.4.2 ASYMMETRY 180 10.4.3 EXPANSION OF SOURCE AND
DISPLACEMENTS INTO SPHERICAL HARMONICS 181 10.4.4 RIGID BODY SPHEROIDAL
MODES 182 SECTION VI: APPENDICES 11. BASIC PROPERTIES OF MATHEMATICAL
FUNCTIONS 185 11.1 BESSEL FUNCTIONS 185 11.1.1 DIFFERENTIAL EQUATION 185
11.1.2 RECURRENCE RELATIONS 185 VIII CONTENTS 11.1.3 DERIVATIVES 186
11.1.4 WRONSKIANS 186 11.1.5 ORTHOGONALITY CONDITIONS 186 11.1.6 USEFUL
INTEGRALS 187 11.2 SPHERICAL BESSEL FUNCTIONS 187 11.2.1 DIFFERENTIAL
EQUATION 187 11.2.2 TRIGONOMETRIC REPRESENTATIONS 188 11.2.3 RECURRENCE
RELATIONS 189 11.3 LEGPNDRE POLYNOMIALS 189 11 J.I DIFFERENTIAL EQUATION
189 11.3.2 RODRIGUES S FORMULA 190 11.3.3 TRIGONOMETRIC EXPANSION (X =
COS 4 ) 19 0 11.3.4 RECURRENCE RELATIONS 190 11.3.5 ORTHOGONALITY
CONDITION 190 11.3.6 EXPANSION IN LEGENDRE SERIES 191 11.4 ASSOCIATED
LEGENDRE FUNCTIONS (SPHEROIDAL HARMONICS) 191 11.4.1 DIFFERENTIAL
EQUATION 191 11.4.2 RECURRENCE RELATIONS 192 11.4.3 ORTHOGONALITY
CONDITIONS 192 11.4.4 ORTHOGONALITY OF CO-LATITUDE MATRIX 193 11.4.5
EXPANSION OF ARBITRARY FUNCTION IN SPHEROIDAL HARMONICS 193 11.4.6
LEIBNIZ RULE FOR THE DERIVATIVE OF A PRODUCT OF TWO FUNCTIONS 193 12.
BRIEF LISTING OF INTEGRAL TRANSFORMS 194 12.1 FOURIER TRANSFORMS 194
12.2 HANKEL TRANSFORMS 196 12.3 SPHERICAL HANKEL TRANSFORMS 197 13.
MATLAB PROGRAMS 198
|
| adam_txt |
FUNDAMENTAL SOLUTIONS IN ELASTODYNAMICS A COMPENDIUM EDUARDO KAUSEL
MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE UNIVERSITY PRESS
CONTENTS PREFACE PAGE IX SECTION I: PRELIMINARIES 1. FUNDAMENTALS 1 1.1
NOTATION AND TABLE OF SYMBOLS 1 1.2 SIGN CONVENTION 4 1.3 COORDINATE
SYSTEMS AND DIFFERENTIAL OPERATORS 4 1.3.1 CARTESIAN COORDINATES 4 1.3.2
CYLINDRICAL COORDINATES 6 1.3.3 SPHERICAL COORDINATES 9 1.4 STRAINS,
STRESSES, AND THE ELASTIC WAVE EQUATION 13 1.4.1 CARTESIAN COORDINATES
13 1.4.2 CYLINDRICAL COORDINATES 16 1.4.3 SPHERICAL COORDINATES 21 2.
DIPOLES 27 2.1 POINT DIPOLES OR DOUBLETS: SINGLE COUPLES AND TENSILE
CRACK SOURCES 27 2.2 LINE DIPOLES 29 2.3 TORSIONAL POINT SOURCES 30 2.4
SEISMIC MOMENTS (DOUBLE COUPLES WITH NO NET RESULTANT) 30 2.5 BLAST
LOADS (EXPLOSIVE LINE AND POINT SOURCES) 31 2.6 DIPOLES IN CYLINDRICAL
COORDINATES 32 SECTION II: FULL SPACE PROBLEMS 3. TWO-DIMENSIONAL
PROBLEMS IN FULL, HOMOGENEOUS SPACES 35 3.1 FUNDAMENTAL IDENTITIES AND
DEFINITIONS 35 3.2 ANTI-PLANE LINE LOAD (SH WAVES) 35 3.3 SH LINE LOAD
IN AN ORTHOTROPIC SPACE 36 3.4 IN-PLANE LINE LOAD (SV-P WAVES) 38 3.6
LINE BLAST SOURCE: SUDDENLY APPLIED PRESSURE 43 3.7 CYLINDRICAL CAVITY
SUBJECTED TO PULSATING PRESSURE 44 4. THREE-DIMENSIONAL PROBLEMS IN
FULL, HOMOGENEOUS SPACES 48 4.1 FUNDAMENTAL IDENTITIES AND DEFINITIONS
48 4.2 POINT LOAD (STOKES PROBLEM) 48 4.3 TENSION CRACKS 52 4.4 DOUBLE
COUPLES (SEISMIC MOMENTS) 54 4.5 TORSFONAL POINT SOURCE 55 4.6 TORSIONAL
POINT SOURCE WITH VERTICAL AXIS 57 4.7 POINT BLAST SOURCE 58 4.8
SPHERICAL CAVITY SUBJECTED TO ARBITRARY PRESSURE 59 4.9 SPATIALLY
HARMONIC LINE SOURCE (2V 2 -D PROBLEM) 63 SECTION III: HALF-SPACE
PROBLEMS 5. TWO-DIMENSIONAL PROBLEMS IN HOMOGENEOUS HALF-SPACES 69 5.1
HALF-PLANE, SH LINE SOURCE AND RECEIVER ANYWHERE 69 5.2 SH LINE LOAD IN
AN ORTHOTROPIC HALF-PLANE 70 5.3 HALF-PLANE, SV-P SOURCE AND RECEIVER AT
SURFACE (LAMB'S PROBLEM) 71 5.4 HALF-PLANE, SV-P SOURCE ON SURFACE,
RECEIVER AT INTERIOR, OR VICE VERSA 73 5.5 HALF-PLANE, LINE BLAST LOAD
APPLIED IN ITS INTERIOR (GARVIN'S PROBLEM) 76 6. THREE-DIMENSIONAL
PROBLEMS IN HOMOGENEOUS HALF-SPACES 78 6.1 3-D HALF-SPACE, SUDDENLY
APPLIED VERTICAL POINT LOAD ON ITS SURFACE (PEKERIS-MOONEY'S PROBLEM) 78
6.2 3-D HALF-SPACE, SUDDENLY APPLIED HORIZONTAL POINT LOAD ON ITS
SURFACE (CHAO'S PROBLEM) 81 6.3 3-D HALF-SPACE, BURIED TORSIONAL POINT
SOURCE WITH VERTICAL AXIS 83 SECTION IV: PLATES AND STRATA 7.
TWO-DIMENSIONAL PROBLEMS IN HOMOGENEOUS PLATES AND STRATA 87 7.1 PLATE
SUBJECTED TO SH LINE SOURCE 87 7.1.1 SOLUTION USING THE METHOD OF IMAGES
87 7.1.2 NORMAL MODE SOLUTION 87 7.2 STRATUM SUBJECTED TO SH LINE SOURCE
88 7.2.1 SOLUTION USING THE METHOD OF IMAGES 89 7.2.2 NORMAL MODE
SOLUTION 90 7.3 PLATE WITH MIXED BOUNDARY CONDITIONS SUBJECTED TO SV-P
LINE SOURCE 90 7.3.1 SOLUTION USING THE METHOD OF IMAGES 91 7.3.2 NORMAL
MODE SOLUTION 92 CONTENTS VII SECTION V: ANALYTICAL AND NUMERICAL
METHODS READ ME FIRST 97 8. SOLUTION TO THE HELMHOLTZ AND WAVE EQUATIONS
98 8.1 SUMMARY OF RESULTS 98 8.2 SCALAR HELMHOLTZ EQUATION IN CARTESIAN
COORDINATES 101 8.3 VECTOR HELMHOLTZ EQUATION IN CARTESIAN COORDINATES
102 8.4 ELASTIC WAVE EQUATION IN CARTESIAN COORDINATES 104 8.4.1
HORIZONTALLY STRATIFIED MEDIA, PLANE STRAIN 105 8.5 SCALAR HELMHOLTZ
EQUATION IN CYLINDRICAL COORDINATES 108 8.6 VECTOR HELMHOLTZ EQUATION IN
CYLINDRICAL COORDINATES 109 8.7 ELASTIC WAVE EQUATION IN CYLINDRICAL
COORDINATES 110 8.7.1 HORIZONTALLY STRATIFIED MEDIA 112 8.7.2 RADIALLY
STRATIFIED MEDIA 114 8.8 SCALAR HELMHOLTZ EQUATION IN SPHERICAL
COORDINATES 117 8.9 VECTOR HELMHOLTZ EQUATION IN SPHERICAL COORDINATES
118 8.10 ELASTIC WAVE EQUATION IN SPHERICAL COORDINATES 120 9. INTEGRAL
TRANSFORM METHOD 125 9.1 CARTESIAN COORDINATES 125 9.2 CYLINDRICAL
COORDINATES 130 9.2.1 HORIZONTALLY STRATIFIED MEDIA 130 9.2.2
CYLINDRICALLY STRATIFIED MEDIA 136 9.3 SPHERICAL COORDINATES 137 10.
STIFFNESS MATRIX METHOD FOR LAYERED MEDIA 140 10.1 SUMMARY OF METHOD 141
10.2 STIFFNESS MATRIX METHOD IN CARTESIAN COORDINATES 142 10.2.1
ANALYTIC CONTINUATION IN THE LAYERS 148 10.2.2 NUMERICAL COMPUTATION OF
STIFFNESS MATRICES 150 10.2.3 SUMMARY OF COMPUTATION 151 10.3 STIFFNESS
MATRIX METHOD IN CYLINDRICAL COORDINATES 159 10.3.1 HORIZONTALLY LAYERED
SYSTEM 160 10.3.2 RADIALLY LAYERED SYSTEM 164 10.4 STIFFNESS MATRIX
METHOD FOR LAYERED SPHERES 175 10.4.1 PROPERTIES AND USE OF IMPEDANCE
MATRICES 180 10.4.2 ASYMMETRY 180 10.4.3 EXPANSION OF SOURCE AND
DISPLACEMENTS INTO SPHERICAL HARMONICS 181 10.4.4 RIGID BODY SPHEROIDAL
MODES 182 SECTION VI: APPENDICES 11. BASIC PROPERTIES OF MATHEMATICAL
FUNCTIONS 185 11.1 BESSEL FUNCTIONS 185 11.1.1 DIFFERENTIAL EQUATION 185
11.1.2 RECURRENCE RELATIONS 185 VIII CONTENTS 11.1.3 DERIVATIVES 186
11.1.4 WRONSKIANS 186 11.1.5 ORTHOGONALITY CONDITIONS 186 11.1.6 USEFUL
INTEGRALS 187 11.2 SPHERICAL BESSEL FUNCTIONS 187 11.2.1 DIFFERENTIAL
EQUATION 187 11.2.2 TRIGONOMETRIC REPRESENTATIONS 188 11.2.3 RECURRENCE
RELATIONS 189 11.3 LEGPNDRE POLYNOMIALS 189 11 J.I DIFFERENTIAL EQUATION
189 11.3.2 RODRIGUES'S FORMULA 190 11.3.3 TRIGONOMETRIC EXPANSION (X =
COS 4 ) 19 0 11.3.4 RECURRENCE RELATIONS 190 11.3.5 ORTHOGONALITY
CONDITION 190 11.3.6 EXPANSION IN LEGENDRE SERIES 191 11.4 ASSOCIATED
LEGENDRE FUNCTIONS (SPHEROIDAL HARMONICS) 191 11.4.1 DIFFERENTIAL
EQUATION 191 11.4.2 RECURRENCE RELATIONS 192 11.4.3 ORTHOGONALITY
CONDITIONS 192 11.4.4 ORTHOGONALITY OF CO-LATITUDE MATRIX 193 11.4.5
EXPANSION OF ARBITRARY FUNCTION IN SPHEROIDAL HARMONICS 193 11.4.6
LEIBNIZ RULE FOR THE DERIVATIVE OF A PRODUCT OF TWO FUNCTIONS 193 12.
BRIEF LISTING OF INTEGRAL TRANSFORMS 194 12.1 FOURIER TRANSFORMS 194
12.2 HANKEL TRANSFORMS 196 12.3 SPHERICAL HANKEL TRANSFORMS 197 13.
MATLAB PROGRAMS 198 |
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| author | Kausel, Eduardo |
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| ctrlnum | (OCoLC)61130818 (DE-599)BVBBV021664647 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 531 - Classical mechanics |
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| discipline | Physik |
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| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV021664647 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:07:00Z |
| indexdate | 2024-07-09T20:41:08Z |
| institution | BVB |
| isbn | 9780521855709 0521855705 |
| language | English |
| lccn | 2005020866 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014879105 |
| oclc_num | 61130818 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-703 DE-706 DE-83 DE-29T |
| owner_facet | DE-91G DE-BY-TUM DE-703 DE-706 DE-83 DE-29T |
| physical | X, 251 S. graph. Darst. 27 cm |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Cambridge University Press |
| record_format | marc |
| spelling | Kausel, Eduardo Verfasser aut Fundamental solutions in elastodynamics a compendium Eduardo Kausel 1. publ. New York Cambridge University Press 2006 X, 251 S. graph. Darst. 27 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references Elasticity Dynamics Green's functions Elastodynamik (DE-588)4151692-8 gnd rswk-swf Green-Funktion (DE-588)4158123-4 gnd rswk-swf Elastodynamik (DE-588)4151692-8 s Green-Funktion (DE-588)4158123-4 s DE-604 http://www.loc.gov/catdir/toc/ecip0516/2005020866.html Table of contents http://www.loc.gov/catdir/enhancements/fy0659/2005020866-d.html Publisher description HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014879105&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kausel, Eduardo Fundamental solutions in elastodynamics a compendium Elasticity Dynamics Green's functions Elastodynamik (DE-588)4151692-8 gnd Green-Funktion (DE-588)4158123-4 gnd |
| subject_GND | (DE-588)4151692-8 (DE-588)4158123-4 |
| title | Fundamental solutions in elastodynamics a compendium |
| title_auth | Fundamental solutions in elastodynamics a compendium |
| title_exact_search | Fundamental solutions in elastodynamics a compendium |
| title_exact_search_txtP | Fundamental solutions in elastodynamics a compendium |
| title_full | Fundamental solutions in elastodynamics a compendium Eduardo Kausel |
| title_fullStr | Fundamental solutions in elastodynamics a compendium Eduardo Kausel |
| title_full_unstemmed | Fundamental solutions in elastodynamics a compendium Eduardo Kausel |
| title_short | Fundamental solutions in elastodynamics |
| title_sort | fundamental solutions in elastodynamics a compendium |
| title_sub | a compendium |
| topic | Elasticity Dynamics Green's functions Elastodynamik (DE-588)4151692-8 gnd Green-Funktion (DE-588)4158123-4 gnd |
| topic_facet | Elasticity Dynamics Green's functions Elastodynamik Green-Funktion |
| url | http://www.loc.gov/catdir/toc/ecip0516/2005020866.html http://www.loc.gov/catdir/enhancements/fy0659/2005020866-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014879105&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kauseleduardo fundamentalsolutionsinelastodynamicsacompendium |